Theorem 2omotap 7477

Description: If there is at most one tight apartness on 2_o, excluded middle follows. Based on online discussions by Tom de Jong, Andrew W Swan, and Martin Escardo. (Contributed by Jim Kingdon, 6-Feb-2025.)

Assertion

Ref	Expression
2omotap	⊢ (∃*𝑟 𝑟 TAp 2o → EXMID)

Proof of Theorem 2omotap

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		2omotaplemst 7476	. . . . 5 ⊢ ((∃*𝑟 𝑟 TAp 2o ∧ ¬ ¬ 𝑥 = {∅}) → 𝑥 = {∅})
2	1	ex 115	. . . 4 ⊢ (∃*𝑟 𝑟 TAp 2o → (¬ ¬ 𝑥 = {∅} → 𝑥 = {∅}))
3		df-stab 838	. . . 4 ⊢ (STAB 𝑥 = {∅} ↔ (¬ ¬ 𝑥 = {∅} → 𝑥 = {∅}))
4	2, 3	sylibr 134	. . 3 ⊢ (∃*𝑟 𝑟 TAp 2o → STAB 𝑥 = {∅})
5	4	adantr 276	. 2 ⊢ ((∃*𝑟 𝑟 TAp 2o ∧ 𝑥 ⊆ {∅}) → STAB 𝑥 = {∅})
6	5	exmid1stab 4298	1 ⊢ (∃*𝑟 𝑟 TAp 2o → EXMID)

Syntax hints:  ¬ wn 3   → wi 4  STAB wstab 837   = wceq 1397  ∃*wmo 2080   ⊆ wss 3200  ∅c0 3494  {csn 3669  EXMIDwem 4284  2_oc2o 6575   TAp wtap 7467

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-stab 838  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-tr 4188  df-exmid 4285  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-1o 6581  df-2o 6582  df-pap 7466  df-tap 7468

This theorem is referenced by:  exmidmotap  7479